Problem: Rewrite as a simplified fraction. ${0.\overline{20} = {?}}$
Solution: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}100x &= 20.202...\\ 1x &= 0.202...\end{align*}}$ Subtracting the two equations, we have: ${99x = 20}$ ${x = \dfrac{20}{99}} $